# What is the largest number you can think of?

This morning we’re going to stretch your minds. What is the largest number you can think of? Is it “120” or “10”? It can’t be “89,000” as that “isn’t even a real number”. These may be the answers may be from school children but it does raise the question: is there really a biggest number?

Infinity you say – solved! Surely infinity is the largest number by its definition. However, this special number is more of an enigma than you might think.

Let’s think about infinity for a moment. First of all, think of numbers 1,2,3,4,5,6 and so on. This series goes on forever; it has a length of infinity. Now pick out just the even numbers: 2,4,6,8 etc and you’d think that now you a list of numbers that contains half the numbers of the infinity list. Wrong. Here we have a paradox of infinity – both lists, the ‘1,2,3’ list and the evens ‘2,4,6’ list, contain the same number of numbers. Both are the same size, both lists are infinitely long. It makes my brain hurt. In the meantime take a breather with a little fact:

It’ll take you about 30 years to count to a billion. And to count from one to a trillion isn’t even possible in the length of human history.

This number infinity has puzzling properties. For example, infinity plus one is infinity. Infinity plus two is infinity and you guessed it, infinity plus infinity equals infinity too. In fact, anything added to infinity is infinity. However, and this is baffling, infinity minus infinity can be anything you like, it could be 1, 2, or 34. Baffling?

Let’s look at an example and say infinity minus infinity is zero. This can be written as:

$\infty - \infty = 0$

Let’s add 34 to each side:

$\infty - \infty + 34 = 0 + 34$

However, $\infty$ added to any number is also $\infty$ so we can simplify the above equation to:

$\infty - \infty = 34$

So suddenly we have two, valid equations saying $\infty - \infty = 34$ and $\infty - \infty = 0$ Puzzling. What this means is that $\infty$ is undefined – you can’t take something away from infinity. Hope your mind has been mentally stretched and limbered up for the week ahead.